Hoàng-Reed Conjecture
Conjecture Every digraph in which each vertex has outdegree at least contains directed cycles such that meets in at most one vertex, .
This conjecture is not even known to be true for . In the case , Thomassen proved [T] that every digraph with minimum outdegree 2 has two directed cycles intersecting on a vertex.
This conjecture would imply the Caccetta-Häggkvist Conjecture.
Bibliography
*[HR] C.T. Hoàng and B. Reed, A note on short cycles in digraphs, Discrete Math., 66 (1987), 103-107.
[T] C. Thomassen, The 2-linkage problem for acyclic digraphs, Discrete Math., 55 (1985), 73-87.
* indicates original appearance(s) of problem.